If Mycroft were the prime minister, I can certainly believe Sherlock wouldn't know unless someone told him. I could just hear them speaking the dialogue. They sounded very in character. And I loved this line as well:

“But it Mycroft Holmes is his real name then that means that he’d have to have constructed a whole alternate personal just to get involved with politics,” John protested. He stopped. “Actually, that does sound like exactly something that you brother would do.”

XD
You know, Mycroft either as the Prime Minister or as the true power behind the prime minister makes lots of sense…
And did I mention you did a fantastic job with the dialogue between John and Sherlock?